1. **Stating the problem:** We will cover the key concepts of the Circle topic for class 10, including definitions, formulas, and example questions similar to previous year board exams. 2. **Definition of a Circle:** A circle is the set of all points in a plane that are at a fixed distance called the radius ($r$) from a fixed point called the center ($O$). 3. **Important terms:** - Radius ($r$): Distance from center to any point on the circle. - Diameter ($d$): Twice the radius, $d = 2r$. - Chord: A line segment joining two points on the circle. - Tangent: A line that touches the circle at exactly one point. - Secant: A line that intersects the circle at two points. 4. **Standard equation of a circle:** If the center is at $(h, k)$ and radius is $r$, the equation is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ If the center is at the origin $(0,0)$, the equation simplifies to: $$ x^2 + y^2 = r^2 $$ 5. **Properties and formulas:** - Length of chord $= 2\sqrt{r^2 - d^2}$ where $d$ is the distance from center to chord. - Equation of tangent to circle at point $(x_1, y_1)$ on circle: $$ xx_1 + yy_1 = r^2 $$ - Area of circle $= \pi r^2$ - Circumference of circle $= 2\pi r$ 6. **Example question from previous year board exam:** *Find the equation of a circle with center at $(3, -2)$ and radius 5.* **Solution:** Using the formula: $$ (x - h)^2 + (y - k)^2 = r^2 $$ Substitute $h=3$, $k=-2$, $r=5$: $$ (x - 3)^2 + (y + 2)^2 = 25 $$ 7. **Another example:** *Find the radius of a circle whose equation is $x^2 + y^2 - 6x + 8y + 9 = 0$.* **Solution:** Rewrite the equation in standard form by completing the square: $$ x^2 - 6x + y^2 + 8y = -9 $$ Complete the square: $$ (x^2 - 6x + 9) + (y^2 + 8y + 16) = -9 + 9 + 16 $$ $$ (x - 3)^2 + (y + 4)^2 = 16 $$ Radius $r = \sqrt{16} = 4$ 8. **Summary:** - Understand the standard form of circle equation. - Practice completing the square to find center and radius. - Use formulas for chords, tangents, and areas. This completes the comprehensive notes and example problems for the Circle topic in class 10.